iQuantum groups and iHopf algebras I: foundation

TL;DR

This paper introduces iHopf algebras, a new algebraic structure on Hopf algebras with pairings, linking quantum groups, braid group actions, and paving the way for constructing dual canonical bases.

Contribution

It defines iHopf algebras, demonstrates their application to quantum groups, and connects them to braid group actions, offering a new framework for quantum algebra research.

Findings

iHopf algebra structure on Borel quantum groups with twisted pairings

Realization of Drinfeld double quantum group as an iHopf algebra

Connections established between Lusztig's braid group action and i-braid group action

Abstract

We introduce the notion of iHopf algebra, a new associative algebra structure defined on a Hopf algebra equipped with a Hopf pairing. The iHopf algebra on a Borel quantum group endowed with a $\tau$-twisted Hopf pairing is shown to be a quasi-split universal iquantum group. In particular, the Drinfeld double quantum group is realized as the iHopf algebra on the double Borel. This iHopf approach allows us to develop connections between Lusztig's braid group action and ibraid group action. It will further lead to the construction of dual canonical basis in a sequel.